Differential Geometry | Topology Answers

Prove that a space curve is a helix if and only if the ratio of the curvature to the torsion is constant at a points

Use the Bisection method to find solutions accurate to within 10^−5 for the following problems. a. x − 2^−x = 0 for 0 ≤ x ≤ 1

b. ex − x^2 + 3x − 2 = 0 for 0 ≤ x ≤ 1

c. 2x cos(2x) − (x + 1)^2 = 0 for −3 ≤ x ≤ −2 and −1 ≤ x ≤ 0 d. x cos x − 2x^2 + 3x − 1 = 0 for 0.2 ≤ x ≤ 0.3 and 1.2 ≤ x ≤ 1.3

Prove that a space curve is a helix if and only if the ratio of the curvature to the torsion is constant at a points

if P is any point on the circle C in the XY plane of a radius a>0 and centre(0,a ) ,let the straight line through origin and P intersect the line y=2a at Q and let the line trough P parallel to the Xaxis intersect the line through Q parallel to the Y axis at R.as P moves around C , R trace out a curve find its paramatrisation and its cartesian equation

find the largest area of the isosceles triangle OAB as shown in the figure below . the point a is on the the x-axis while the point b lies on the curve y=49-x² in the first quadrant( use second derivative method)